## Prices and Demands (Part II)

In part one we have seen that in microeconomics if we treat demand and price as observables on a Hilbert space, then not both of them can be bounded linear operators. Especially, since all linear operators on a finite dimensional Hilbert space are bounded, the state space for our market cannot be finite dimensional. All this is a consequence of a single assertion, namely demand invariance under price-scaling and somehow resembles the situation in quantum theory.

So far our considerations concerning prices and demands were quite abstract. To do some real computations we need a representation of these concepts. As promised last week, I now provide such a representation for the observables price ${p_i}$, demand ${d_i}$ and excess demand ${z_i}$ on an appropriate Hilbert space.

The results so far, lead to the following approach: The Hilbert space is given as ${X = L^2(\mathbb{R}^n)}$. This, in a way, is the simplest non-finite dimensional Hilbert space and therefore an unsurprising first choice. Thus, the state of the market with ${n\in\mathbb{N}}$ goods is described by a function ${\xi\in X}$. Let the coordinates of ${\xi}$ be denoted as ${(x_1,\ldots,x_n)\in \mathbb{R}^n}$, then the demand ${d_i: D(d_i)\rightarrow X}$ is given as a differential operator

$\displaystyle d_i \xi = -i \mu_i \frac{d}{d x_i} \xi$

with domain

$\displaystyle D(d_i) = \{ \xi\in X: \xi \textnormal{ absolutely continuous and } \xi' \in X \}.$

The excess demand operator ${z_i=d_i-\omega_i \textnormal{id}_X}$ has the same domain as ${d_i}$.
Define the function ${e_i:\mathbb{R}^n\rightarrow \mathbb{R}}$ as ${e_i(x)=e^{x_i}}$. Then, the price operator ${p_i:D(p_i)\rightarrow X}$ is given as a multiplication operator

$\displaystyle p_i \xi = e_i \cdot \xi$

with domain

$\displaystyle D(p_i) = \{ \xi\in X: e_i \cdot \xi \in X \}.$

All operators ${p_i, d_i, z_i}$ are self-adjoint, ${p_i}$ is positive, the commutator satisfies

$\displaystyle \left[p_i,z_i\right]\xi = \left[p_i,d_i\right]\xi = -i \mu_i \left(e_i \cdot \frac{d}{d x_i}\xi - e_i \cdot \xi - e_i \cdot \frac{d}{d x_i} \xi\right)= i \mu_i p_i \xi$

and thus the market axioms are fulfilled.

That still might look a little abstract if one is not used to Functional Analysis. The corresponding representation in quantum mechanics however lead to major new insights into the field.

Next time I shall give a formal comparison of microeconomics and quantum mechanics. As you can imagine by now, they are similar on some abstract level. However, there are also some striking discrepancies like the different commutation relations and thus the step towards the desired Schrödinger type equation for markets is not straight forward.

Stay tuned …